3.3 Observation Volume in Organic Solvents
The dye labelled polystyrenes were used to measure the diffusion of single chains by FCS.
The experiments were conducted on the modified FCS setup described in chapter 2.2. For the measurement the Rhodamine B label of the chain was excited by an Ar-ion laser at 514 nm. The fluorescence light was separated from scattered light by a 514 nm beamsplitter.
An additional long-pass filter with a cut-off wavelength of 560 nm was used to suppress the remaining scattered excitation light. For every measurement the polymer solutions were freshly prepared with toluene p.a. grade and filled in the sealed sample chamber described in section 2.4.1. Three autocorrelation curves with a recording time of 5 min were measured for each molecular weight.
Figure 3.4 shows the curves obtained for 10−8M polymer solutions of four different
0 . 0 1 0 . 1 1 1 0 1 0 0 1 0 0 0
Figure 3.4: Normalized FCS autocorrelation curves for 10−8M solutions of different la-belled polystyrenes in toluene. The molecular weight of the polystyrenes is from left to right: 11.5k, 19k, 63k, and 285k.
molecular weights. For increasing molecular weight the autocorrelation curves show a clear increase in the diffusion time. The diffusion times τDiff were extracted by a fit of equation 2.21 to the data. The obtained diffusion times were plotted versus the molecular weight
1 0 k 1 0 0 k 1 M 1 E - 4
Diffusion time [s]
M o l e c u l a r w e i g h t [ g / m o l ]
Figure 3.5: Single molecule diffusion times as a function of molecular weight as determined from least squares fits to the data. The straight line shows a linear fit to the data.
(figure 3.5). We recall that the diffusion times τDiff are correlated to the waist radius by τDiff = 4Dw2 as described in equation 2.16. A double-logarithmic presentation of our experimental data shown in figure 3.5 reveals a power law dependence between molecular weight and diffusion time. The fitting procedure yields the exponent and prefactor of the diffusion time – molecular weight relationship.
τ(Mw) =k·Mwa ⇒log τ(Mw) = (−6.050±0.021) + (0.498±0.0043)·log Mw (3.1) D(Mw) = w2
4τ(Mw) = w2
4k ·Mw−a⇒D(Mw) = w2
4·8.912·10−7 ·Mw−0.498 (3.2) Bugdahl studied the molecular weight dependence of the diffusion coefficient for polystyrene in toluene by dynamic light scattering [Bug69] and found the following scaling law:
D(Mw) = (2.15·10−4)·Mw−0.53=b·Mw−a (3.3)
3.3. OBSERVATION VOLUME IN ORGANIC SOLVENTS 41 By comparison of the prefactors of the equations 3.2 and 3.3 the waist radius of the obser-vation volume in toluene can be extracted.
b = w2
4·8.912·10−7 ⇒w=
√
4·8.912·10−7·b= 277 nm (3.4)
For toluene a waist radius ofwtoluene = 277 nm was obtained. This value differs significantly from the value of the waist radius in water, which was determined to bewwater= 212±4 nm by the Rhodamine 6G measurements described earlier.
As outlined in chapter 2.3 theoretical considerations of the focus size can only be done by exact wave-optical calculations. In figure 3.6 the calculated MDF is shown and the measured waist radius of our observation volume is represented by a black bar. It is clearly seen that the region of high intensity from the calculated MDF and the measured waist radius are identical. Only with the new synthesized polymer system for the determination of the observation volume we were able to measure the size of the observation volume precisely and achieve a high agreement with the theoretical calculated size of the focus volume.
In summary, a procedure was developed to determine reliably the size of the observation volume in organic solvents with different refractive indices. Therefore polymers with a single dye molecule coupled to the end of the chain were synthesised and characterised by MALDI-ToF and GPC-measurements. To achieve narrow distributions in molecular weight all polymers were synthesised by anionic polymerisation. This fact and the use of five different molecular weights in a range of 10 to 1550kg/mol enable FCS measurements with high validity. Hence a waist radius of the focus volume in toluene was received which is in excellent agreement with calculations based on wave-optical considerations. The successful combination of polymer synthesis and modifications of the FCS setup enable high quality FCS measurements in organic solutions.
1
-1 0
0 1 2
-1 -2
Figure 3.6: Calculated intensity distribution for an objective with a numerical aperture of 0.9 used for measurements in toluene. The black line indicates the waist radius wxy deter-mined from measurements with dye labelled polystyrenes of different molecular weights.
Chapter 4
Determination of the Crossover between Dilute and Semi-Dilute Polymer Solutions
Polymer solutions constitute an important part of polymer science. Often synthesis, char-acterisation and processing of polymers take place in solutions of different concentrations.
In some cases like synthesis and spin casting the polymer concentration varies in a broad range during the experiment. Phase diagrams of polymer solutions yield information about the relation between concentration, molecular weight, temperature and composition of polymers. Variation of the parameters leads to changes in the physical properties of the solution. Phase diagrams which cover a broad range in molecular weight and concentration can so far only be measured by using various methods for the investigation of the physical properties of the polymer solutions in the different concentration regimes and for the dif-ferent molecular weights. Thus, it will be very helpful to devise an experimental method which covers the full range of concentration regimes for largely difficult molecular weights.
Single molecule methods often use tracer molecules to investigate the mobility of single 43
nal to background ratio compared to DLS circumvents the restrictions in molecular weight.
The high sensitivity of FCS for example on the change in the mobility of molecules allow to follow the onset of the beginning of the overlap of polymer chains with increasing concen-tration. The starting point of the overlap of polymer chains is localized in a concentration range which is hardly fully accessible by classical methods.
In the previous chapters we described how to modify a standard FCS setup for measure-ments in polymer solutions. It was shown by Liu et al [Liu05] that this technique allows to measure the mobility of single polymer chains in a broad concentration range.We use this advanced method to investigate polystyrene solutions within a broad range of concen-trations and different molecular weights. These experiments will yield information about mobility changes of single polymer chains at the crossover between dilute and semi-dilute polymer concentrations.
4.1 Dilute and Semi-Dilute Polymer Solutions
Polymer solutions are categorized in three different concentration regimes. In phase dia-grams dilute, semi-dilute and concentrated polymer concentrations are distinguished. The locus of the crossover between the different regimes depends on the concentration and on the chain dimensions. In the following chapter we will concentrate on the dilute and semi-dilute regime and especially on the crossover between these two regimes.
At very high dilution the single polymer chains move separately from each other and do not show any interaction. This regime is called the dilute state. With increasing poly-mer concentration the chains start to touch each other. This concentration is called the overlap concentration (c∗). By increasing the polymer concentration even more the poly-mer chains start to interpenetrate (figure 4.1) and the semi-dilute regime is reached. The
4.1. DILUTE AND SEMI-DILUTE POLYMER SOLUTIONS 45 overlap concentration usually is not a sharp transition due to polydispersity. It refers to a concentration range between the dilute and semi-dilute region. For the understanding of polymer solutions it is very useful to investigate the scaling properties ofc∗with the molec-ular weight. The basic definition of the overlap concentration is given by de Gennes [Gen79]
and by Strobl [Str97]: The monomer concentration in the solution has to be the same as the monomer concentration in the volume occupied by a single polymer coil. Though the
c=c* c>c*
c<c*
Figure 4.1: Concentration regimes of polymer solutions. Dilute regime c < c∗ and semi-dilute regime c > c∗. In the semi-dilute regime the polymer chains overlap. The crossover occurs when the volume of the individual chains is the same as the sample volume. At that point the concentration of monomers inside the volume of a single chain is the same as the monomer concentration in the entire polymer solution cm =c∗m.
crossover takes place in a concentration range Strobl suggests for expanded chains an exact concentration defined by:
c∗= N
R3F (4.1)
where N is the degree of polymerisation andRF is the “Flory radius”. This radius is a mea-sure for the volume occupied by a polymer chain in the corresponding solvent. With this expression it is possible to give a scaling law for the dependence of the overlap concentration
F
RF =aF ·Nν (4.2)
aF denotes the effective length of a single monomer unit. The exponent ν describes the shape of the polymer chains. The Flory-Huggins theory gives for chains in a Θ-solvent ν = 12 and for coiled chains in a good solvent ν reaches 35. Inserting equation 4.2 in 4.1 yields the scaling law for the overlap concentration.
c∗= N
a3F ·N3ν = N1−3ν
a3F (4.3)
Using the relation between the degree of polymerisationN and the weight average of the molecular weightMw of the polymer sample and the molecular weight of a monomer unit M:
N = Mw
M (4.4)
the scaling law can be rewritten as:
c∗= 1
a3F ·M1−3ν ·Mw1−3ν (4.5)
It is convenient to define the volume fraction of a polymer in solution.
φ=vm·cm (4.6)
vm is the volume of a monomer. With this definition equation 4.5 reads:
φ∗=vm·cm∗= vm
a3f ·N1−3ν (4.7)
4.1. DILUTE AND SEMI-DILUTE POLYMER SOLUTIONS 47 The concentration range of the overlap concentration is difficult to reach by conventional methods typically used in polymer science. Rheology and scattering methods do not give direct access to the overlap concentration. Moreover rheology measurements always disturb the system by shearing and do not give direct information about the undisturbed system.
For scattering methods there are too many scattering centres in the semi-dilute concen-tration regime. Besides light scattering experiments are restricted to molecular weights higher than 15 kg/mol. To get information about the overlap concentration often geomet-rical considerations are used. Graessley for example measured the intrinsic viscosity of polystyrene solutions. From the achieved values he calculated the overlap concentration of the polymer chains [Gra80]. The results of his calculations are shown in figure 4.2. Other groups calculated the overlap concentration with hydrodynamic radii determined by light scattering experiments. None of these studies based on geometrical calculations measures the overlap concentration directly.
Brown et al. [Bro88] measured the change of the diffusion coefficient for three molecular weights and extracted from the change of the diffusion coefficient the overlap concentration.
The proof of the scaling law given in equation 4.5 was only possible in a range of molecular weight ranging from 100 to 2900kg/mol. Hervet et al. [Her79] used forced Rayleigh light scattering to investigate changes in the mobility of polymers. They investigated only the two molecular weights 123 and 245kg/mol. Hence, they only determined the overlap con-centration and did not proof any scaling law with their results.
A more direct determination of the crossover between the dilute and the semi-dilute regime by investigating the mobility of single polymer chains in a broad range of molec-ular weights would give a more detailed insight in the behaviour of the polymers at this crossover. Single molecule methods give the possibility to avoid all these restrictions men-tioned above. Working with highly diluted tracer molecules allows the direct observation of the behaviour of single polymer chains. From the change of the mobility of single polymer
1k
Figure 4.2: Concentration versus molecular–weight for polystyrene in a good solvent. The crossover between the dilute and the semi-dilute concentration was calculated from intrinsic viscosity data by Graessley [Gra80].
chains the overlap concentration can directly extracted.
4.2 Determination of the Overlap Concentration of Polystyrene with FCS
We want to determine the overlap concentration of polystyrene dissolved in toluene. For the experiments we used the same polymers as for the analysis of the shape and size of the ob-servation volume (chapter 3.2). For the experiments solutions of dye-labelled polystyrenes of a concentration of 10−8M were prepared. For the measurement a low dye concentration is needed (see theory chapter 2.1). To vary the polymer concentration and keep the dye concentration constant unlabelled polymer from the same synthesis batch is added to the